diff options
Diffstat (limited to 'src/algebra/cycles.spad.pamphlet')
| -rw-r--r-- | src/algebra/cycles.spad.pamphlet | 36 |
1 files changed, 15 insertions, 21 deletions
diff --git a/src/algebra/cycles.spad.pamphlet b/src/algebra/cycles.spad.pamphlet index 4fca9d13..b8bf2e16 100644 --- a/src/algebra/cycles.spad.pamphlet +++ b/src/algebra/cycles.spad.pamphlet @@ -32,35 +32,36 @@ CycleIndicators: Exports == Implementation where PTN ==> Partition RN ==> Fraction Integer FR ==> Factored Integer + macro NNI == NonNegativeInteger Exports ==> with - complete: I -> SPOL RN + complete: NNI -> SPOL RN ++\spad{complete n} is the \spad{n} th complete homogeneous ++ symmetric function expressed in terms of power sums. ++ Alternatively it is the cycle index of the symmetric ++ group of degree n. - powerSum: I -> SPOL RN + powerSum: NNI -> SPOL RN ++\spad{powerSum n} is the \spad{n} th power sum symmetric ++ function. - elementary: I -> SPOL RN + elementary: NNI -> SPOL RN ++\spad{elementary n} is the \spad{n} th elementary symmetric ++ function expressed in terms of power sums. - alternating: I -> SPOL RN + alternating: NNI -> SPOL RN ++\spad{alternating n} is the cycle index of the ++ alternating group of degree n. - cyclic: I -> SPOL RN --cyclic group + cyclic: NNI -> SPOL RN --cyclic group ++\spad{cyclic n} is the cycle index of the ++ cyclic group of degree n. - dihedral: I -> SPOL RN --dihedral group + dihedral: NNI -> SPOL RN --dihedral group ++\spad{dihedral n} is the cycle index of the ++ dihedral group of degree n. - graphs: I -> SPOL RN + graphs: NNI -> SPOL RN ++\spad{graphs n} is the cycle index of the group induced on ++ the edges of a graph by applying the symmetric function to the ++ n nodes. @@ -102,12 +103,8 @@ CycleIndicators: Exports == Implementation where list st == entries complete st complete i == - if i=0 - then 1 - else if i<0 - then 0 - else - _+/[trm(partition pt) for pt in list(partitions i)] + i=0 => 1 + +/[trm(partition pt) for pt in list(partitions i)] even?: L I -> B @@ -117,12 +114,8 @@ CycleIndicators: Exports == Implementation where 2 * _+/[trm(partition li) for li in list(partitions i) | even? li] elementary i == - if i=0 - then 1 - else if i<0 - then 0 - else - +/[(spol := trm(partition pt); even? pt => spol; -spol) + i=0 => 1 + +/[(spol := trm(partition pt); even? pt => spol; -spol) for pt in list(partitions i)] divisors: I -> L I @@ -203,7 +196,8 @@ CycleIndicators: Exports == Implementation where wreath(spol1,spol2) == evspol(mtpol(#1,spol2),spol1) SFunction li== - a:Matrix SPOL RN:=matrix [[complete(k -j+i) for k in li for j in 1..#li] + a:Matrix SPOL RN := + matrix [[complete((k -j+i)::NNI) for k in li for j in 1..#li] for i in 1..#li] determinant a @@ -216,7 +210,7 @@ CycleIndicators: Exports == Implementation where #li1 < #li2 => error "skewSFunction: partition1 does not include partition2" li2:=roundup (li1,li2) - a:Matrix SPOL RN:=matrix [[complete(k-li2.i-j+i) + a:Matrix SPOL RN:=matrix [[complete((k-li2.i-j+i)::NNI) for k in li1 for j in 1..#li1] for i in 1..#li1] determinant a |